Mastering 0 Mathematlcs
Macmillan Master Series Accounting Advanced English Language Advanced Pure Mathematics Arabic Banking Basic Management Biology British Politics Business Administration Business Communication Business Law C Programming Catering Theory Chemistry COBOL Programming Communication Databases Economic and Social History Economics Electrical Engineering Electronic and Electrical Calculations Electronics English as a Foreign Language English Grammar English Language English Literature English Spelling French French 2 German German 2 Global Information Systems Human Biology Italian Italian 2 Japanese Manufacturing Marketing Mathematics Mathematics for Electrical and Electronic Engineering Modem British History Modem European History Modem W orld History Pascal Programming Philosophy Photography Physics Psychology Science Social Welfare Sociology Spanish Spanish 2 Spreadsheets Statistics Study Skills Visual Basic W ord Processing Macmillan Master Series Series Standing Order ISBN 978-0-333-69343-8 If you would like to receive future titles in this series as they are published, you can make use of our standing order facility. To place a standing order please contact your bookseller or, in case of difficulty, write to us at the address below with your name and address, the title ofthe series and the ISBN quoted above. (Ifyou live outside the United Kingdom we may not have the rights for your country, in which case we will forward your order to the publisher concemed.) Customer Services Department, Macmillan Distribution Ltd Houndmills, Basingstoke, Hampshire RG2l 6XS, England
o Mastering Mathematics Second Edition Geoff Buckwell ~ MACMlUAN
Geoff Buckwell 1991, 1997 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London WIP 9HE. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First edition 1991 Reprinted four times Second edition 1997 Published by MACMILLAN PRESS LTD Houndmills, Basingstoke, Hampshire RG21 6XS and London Companies and representatives throughout the world ISBN 978-0-333-66508-4 ISBN 978-1-349-14131-9 (ebook) DOI 10.1007/978-1-349-14131-9 A catalogue record for this book is available from the British Library. This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. 10 9 8 7 6 5 06 05 04 03 02 01 432 1 00 99 98 97 Copy-edited and typeset by Povey-Edmondson Tavistock and Rochdale, England
Ocontents Introduction Acknowledgements 1 Working with whole numbers 1.1 Number systems 1.2 The four ru1es of working with numbers 1.3 The use of brackets 1.4 Multiples and factors 1.5 Negative numbers 1.6 Other number bases 2 Working with fractions and decimals 2.1 U nderstanding fractions 2.2 Decima1 fractions 2.3 Decimal places and significant figures 2.4 Standard index or scientific notation 2.5 Working with decima1s 2.6 Squares and square roots 2.7 Approximation and the calculator 2.8 The metric system 2.9 Speed, distance, time 3 Introducing algebra 3.1 Basic definitions 3.2 Formulae and substitution 3.3 Forming expressions 3.4 Simplifying expressions 3.5 Equations (linear) 3.6 Problem solving with equations 3.7 Rearranging formulae (changing the subject) 4 Ratios and percentages 4.1 Percentages 4.2 Interest 4.3 Ratio and proportion 4.4 Changing units and currency conversion 5 Angle and shape 5.1 Definitions 5.2 Polygons 5.3 Symmetry of polygons IX X 1 1 3 8 10 14 20 27 27 39 41 42 45 48 50 51 54 60 60 60 62 64 67 72 73 76 76 83 86 90 97 97 103 106 CONTENTS v
5.4 Properties of three and four sided shapes 108 5.5 Similarity and congruence 113 5.6 Tessellations 117 6 Length, area and volume 122 6.1 Perimeter of plane figures 122 6.2 Area 125 6.3 Nets and surface area 133 6.4 Volume 137 6.5 Similar shapes 142 7 Trigonometry 146 7.1 Tangent ratio 146 7.2 Sine ratio 150 7.3 Cosine ratio 154 7.4 Angle of elevation (or depression), bearings 158 7.5 Pythagoras' theorem 163 7.6 Three-dimensional problems 170 8 Geometry and construction 175 8.1 Geometrical constructions 175 8.2 Drawing three-dimensional shapes 182 8.3 Plans and elevations 183 8.4 Latitude and longitude 186 8.5 Locus 191 9 Statistics 194 9.1 Statistical data 194 9.2 Statistical diagrams 195 9.3 Statistical averages 203 9.4 Cumulative frequency 208 9.5 Scatter dia grams 214 10 More algebra 219 10.1 Fractional and negative indices 219 10.2 Algebraic fractions 222 10.3 Quadratic expressions 223 10.4 Number patterns 225 10.5 Factorisation 228 10.6 Quadratic equations 232 10.7 Trial and improvement 235 10.8 Problem solving 236 10.9 Further rearrangement of formulae 240 10.10 Simultaneous equations (linear) 244 10.11 Variation and proportion 246 10.12 Venn diagrams 249 11 Probability 254 11.1 Definition 254 11.2 Probability from graphs 257 vi CONTENTS
11.3 Using Venn diagrarns 11.4 Surn and produet rules 11.5 Complement 11.6 Tree diagrams 12 Coordinates and straight line graphs 12.1 Using straight line graphs 12.2 Coordinates and the equation of a straight line 12.3 y = mx + c 12.4 Simultaneous equations 12.5 Trave1 graphs 12.6 Experimentallaws 12.7 Inequalities 12.8 Errors 12.9 Regions and linear programming 13 Curves, functions and equations 13.1 Plotting eurves 13.2 Solving equations 13.3 The trapezium rule 13.4 Funetion notation 13.5 Flow charts 13.6 Iterative methods 14 The circle and further trigonometry 14.1 Geometry of the eircle 14.2 The sine rule 14.3 The eosine rule 14.4 Angles greater than 90 14.5 Trigonometrie equations 15 Matrices 15.1 Basic definitions 15.2 Route matriees 15.3 Algebra of matriees 16 Vectors and transformations 16.1 Column vectors 16.2 Geometrieal proofs 16.3 Transformations 17 The calculus 17.1 Basic ideas of differentiation 17.2 Maxima and minima 17.3 Integration 17.4 Area and volume Hints and solutions to investigations Answers to exercises Index 258 261 262 265 272 272 273 276 278 282 288 292 294 295 300 300 302 305 311 316 318 322 322 334 339 343 344 348 348 351 356 363 363 368 372 382 382 388 389 391 395 399 423 CONTENTS vii
o Introduction This book covers basic mathematical principles in a clear, straightforward style. The purpose of this book is to help those of you who are finding some (or all!) aspect of mathematics difficult. Very often, it is a small point that needs explanation, and many books do not cover everything in sufficient detail. Mastering Mathematics is designed to overcome these problems. It includes pencil and paper techniques as weh as the correct use of a calculator. It also covers a wide range of topics so that if you are following a GCSE, BTEC, college course, or just 'doing mathematics for fun', you will almost certainly find the topics you need. Scattered throughout the book are a number of Investigations which are designed to help you explore topics in a more practical way. Also as you progress through the book, there are Examples and Exercises to see if you have indeed mastered those subjects covered in that chapter. None of these investigations requires any specialist equipment. It is important to remember when studying mathematics at any time that methods sometimes need to be read several times. Perseverance will bring its reward, particularly if the exercises are tried and checked against the answers at the back. If you get anything wrong, go back to the text and read the relevant section. Geoff Buckwell INTRODUCTION ix
o Acknowledgements I am indebted to Colin Prior for his illustrations which I hope give some light relief, to Josie Buckwell for her work in preparing and checking the manuscript, and Matthew Buckwell for his work in producing the diagrams. The author and publishers wish to thank the following for permission to use copyright material: Express Newspapers pie for articles 'Quiet for God's Sake' and 'No Camping', Daily Star, 4 March 1991. The London East Anglian Group: Midland Examining Group; Northern Examining Association comprising Associated Lancashire Schools Examining Board, Joint Matriculation Board, North Regional Examinations Board, North West Regional Examinations Board and Yorkshire & Humberside Regional Examinations Board; Southern Examining Group, University of Cambridge Local Examinations Syndicate, and the Welsh Joint Education Committee for questions from past examination papers. Times Newspapers Ltd for 'Britain will honour troops with Gulf victory parade' by Nicholas Wood and Ruth Gledill, The Times, 4 March 1991. Every effort has been made to trace all the copyright-holders, but if any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangement at the first opportunity. Geoff Buckwell x ACKNOWLEDGEMENTS